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Maximum modulus, characteristic, and area on the sphere. (English) Zbl 0703.30025
Let f be an entire function of infinite order. T(r) its Ahlfors-Shimizu characteristic. If $$\gamma$$ (r) is a differentiable, increasing function such that T(r)$$\leq \gamma (r)$$ $$(r>r_ 0)$$, then
(a) $$\liminf_{r\to \infty}(\log M(r,f)/r\gamma '(r))\leq \pi;$$
(b) If log $$\gamma$$ (r) is a convex function of log r, $$\limsup_{r\to \infty}(\log M(r,f)/r\gamma '(r))\leq \pi.$$
The choice $$\gamma (r)=T(r)$$ together with known results [N. V. Govorov, Funkts. Anal. Prilozh. 3, No.2, 41-45 (1969; Zbl 0197.053)] yields:
For every entire function of order $$\geq 1/2$$ $\liminf_{r\to \infty}\log M(r,f)/A(r)\leq \pi.$ Here A(r) is the area of the image of $$\{| z| \leq r\}$$ under the map f: $${\mathbb{C}}\to Riemann$$ sphere.
The paper also identifies increasing functions $$\psi$$ such that $\liminf_{r\to \infty}\log M(r,f)/\Psi (T(r,f))\leq \pi.$ The proofs are based on a clever application of real variable results to the Baernstein $$T^*$$ function.
Reviewer: W.H.J.Fuchs

##### MSC:
 30D20 Entire functions of one complex variable, general theory 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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